1. ## Carmichael Number Question

Show that every integer of the form (6m+1)(12m+1)(18m+1), where m is a positive integer such that 6m+1, 12m+1, and 18m+1 are all primes, is a Carmichael number.

2. Originally Posted by Janu42
Show that every integer of the form (6m+1)(12m+1)(18m+1), where m is a positive integer such that 6m+1, 12m+1, and 18m+1 are all primes, is a Carmichael number.
Korselt's criterion

Carmichael number - Wikipedia, the free encyclopedia

3. Originally Posted by undefined

OK I see, we didn't have that in our book. But how do I use this to prove this specific case? I figure it must have something to do with singling out the specific m's so that 6m+1, 12m+1, and 18m+1 are all prime.

4. Originally Posted by Janu42
OK I see, we didn't have that in our book. But how do I use this to prove this specific case? I figure it must have something to do with singling out the specific m's so that 6m+1, 12m+1, and 18m+1 are all prime.
We know (6m+1)(12m+1)(18m+1) has exactly three prime factors, all of them distinct, based on what's given in the problem. So we test each prime in turn.

Show that 6m | (6m+1)(12m+1)(18m+1)-1

you can do so by expanding the right hand side and factoring out 6m from the result.

In fact you can combine all three steps considering that lcm(6m,12m,18m) = m*lcm(6,12,18) = 36m.